{-# OPTIONS --lossy-unification #-}
module Cubical.CW.Approximation where
open import Cubical.CW.Base
open import Cubical.CW.Properties
open import Cubical.CW.Map
open import Cubical.CW.Homotopy
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Function
open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)
open import Cubical.Data.Nat.Order
open import Cubical.Data.Fin.Base
open import Cubical.Data.Fin.Properties
open import Cubical.Data.Sigma
open import Cubical.Data.Unit
open import Cubical.Data.Empty as ⊥
open import Cubical.Data.Sequence
open import Cubical.Data.FinSequence
open import Cubical.Data.Nat.Order.Inductive
open import Cubical.Data.Sum as ⊎
open import Cubical.HITs.SequentialColimit
open import Cubical.HITs.PropositionalTruncation as PT hiding (elimFin)
open import Cubical.HITs.Truncation as TR
open import Cubical.HITs.Sn
open import Cubical.HITs.Pushout
open import Cubical.Axiom.Choice
open import Cubical.Homotopy.Connected
open Sequence
open FinSequenceMap
private
variable
ℓ ℓ' ℓ'' : Level
finCellApprox : (C : CWskel ℓ) (D : CWskel ℓ')
(f : realise C → realise D) (m : ℕ) → Type (ℓ-max ℓ ℓ')
finCellApprox C D f m =
Σ[ ϕ ∈ finCellMap m C D ]
(FinSeqColim→Colim m {X = realiseSeq D}
∘ finCellMap→FinSeqColim C D ϕ
≡ f ∘ FinSeqColim→Colim m {X = realiseSeq C})
idFinCellApprox : (m : ℕ)
{C : CWskel ℓ} → finCellApprox C C (idfun _) m
fst (idFinCellApprox m {C}) = idFinCellMap m C
snd (idFinCellApprox m {C}) =
→FinSeqColimHomotopy _ _ λ x → refl
compFinCellApprox : (m : ℕ)
{C : CWskel ℓ} {D : CWskel ℓ'} {E : CWskel ℓ''}
{g : realise D → realise E}
{f : realise C → realise D}
→ finCellApprox D E g m → finCellApprox C D f m
→ finCellApprox C E (g ∘ f) m
fst (compFinCellApprox m {g = g} {f} (F , p) (G , q)) = composeFinCellMap m F G
snd (compFinCellApprox m {C = C} {g = g} {f} (F , p) (G , q)) =
→FinSeqColimHomotopy _ _ λ x
→ funExt⁻ p _
∙ cong g (funExt⁻ q (fincl _ x))
finCellHomRel : {C : CWskel ℓ} {D : CWskel ℓ'} (m : ℕ)
(f g : finCellMap m C D)
(h∞ : (n : Fin (suc m)) (c : fst C (fst n))
→ Path (realise D) (incl (f .fmap n c)) (incl (g .fmap n c)))
→ Type (ℓ-max ℓ ℓ')
finCellHomRel {C = C} {D = D} m f g h∞ =
Σ[ ϕ ∈ finCellHom m f g ]
((n : Fin (suc m)) (x : fst C (fst n))
→ Square (h∞ n x)
(cong incl (finCellHom.fhom ϕ n x))
(push (f .fmap n x)) (push (g .fmap n x)))
module _ (C : CWskel ℓ) (D : CWskel ℓ') (f : realise C → realise D) where
find-connected-component : (d : realise D) → ∃[ d0 ∈ fst D 1 ] incl d0 ≡ d
find-connected-component = CW→Prop D (λ _ → squash₁) λ a → ∣ a , refl ∣₁
find-connected-component-C₀ : (c : fst C 1)
→ ∃[ d0 ∈ fst D 1 ] incl d0 ≡ f (incl c)
find-connected-component-C₀ c = find-connected-component (f (incl c))
approx₁ : ∃[ f₁ ∈ (fst C 1 → fst D 1) ] ((c : _) → incl (f₁ c) ≡ f (incl c))
approx₁ =
invEq (_ , satAC∃Fin-C0 C (λ _ → fst D 1) (λ c d0 → incl d0 ≡ f (incl c)))
find-connected-component-C₀
approx : (m : ℕ)
→ ∃[ fₙ ∈ ((n : Fin (suc m)) → Σ[ h ∈ (fst C (fst n) → fst D (fst n)) ]
((c : _) → incl (h c) ≡ f (incl c))) ]
((n : Fin m) (c : fst C (fst n))
→ fₙ (fsuc n) .fst (CW↪ C (fst n) c)
≡ CW↪ D (fst n) (fₙ (injectSuc n) .fst c))
approx zero = ∣ (λ { (zero , p)
→ (λ x → ⊥.rec (CW₀-empty C x))
, λ x → ⊥.rec (CW₀-empty C x)})
, (λ {()}) ∣₁
approx (suc zero) =
PT.map (λ a1 →
(λ { (zero , p) → (λ x → ⊥.rec (CW₀-empty C x))
, λ x → ⊥.rec (CW₀-empty C x)
; (suc zero , p) → a1})
, λ {(zero , p) c → ⊥.rec (CW₀-empty C c)})
approx₁
approx (suc (suc m)) =
PT.rec
squash₁
(λ {(p , f')
→ PT.rec squash₁ (λ r
→ PT.map (λ ind → mainₗ p f' r ind
, mainᵣ p f' r ind)
(mere-fib-f-coh (p flast .fst)
r (p (suc m , <ᵗsuc) .snd)))
(fst-lem (p flast .fst)
(p flast .snd))})
(approx (suc m))
where
open CWskel-fields C
fst-lem : (fm : fst C (suc m) → fst D (suc m))
→ ((c : fst C (suc m)) → incl (fm c) ≡ f (incl c))
→ ∥ ((x : A (suc m)) (y : S₊ m)
→ CW↪ D (suc m) (fm (α (suc m) (x , y)))
≡ CW↪ D (suc m) (fm (α (suc m) (x , ptSn m)))) ∥₁
fst-lem fm fh =
invEq propTrunc≃Trunc1
(invEq (_ , InductiveFinSatAC 1 (CWskel-fields.card C (suc m)) _)
λ a → fst propTrunc≃Trunc1
(sphereToTrunc (suc m) λ y →
TR.map fst (isConnectedCong _ _ (isConnected-CW↪∞ (suc (suc m)) D)
(sym (push _)
∙ (fh (CWskel-fields.α C (suc m) (a , y))
∙ cong f (push _
∙ cong incl (cong (invEq (CWskel-fields.e C (suc m)))
(push (a , y) ∙ sym (push (a , ptSn m))))
∙ sym (push _))
∙ sym (fh (CWskel-fields.α C (suc m) (a , ptSn m))))
∙ push _) .fst)))
module _ (fm : fst C (suc m) → fst D (suc m))
(fm-coh : (x : A (suc m)) (y : S₊ m) →
CW↪ D (suc m) (fm (α (suc m) (x , y)))
≡ CW↪ D (suc m) (fm (α (suc m) (x , ptSn m)))) where
module _ (ind : ((c : fst C (suc m)) → incl (fm c) ≡ f (incl c))) where
fib-f-incl : (c : fst C (suc (suc m))) → Type _
fib-f-incl c = Σ[ x ∈ fst D (suc (suc m)) ] incl x ≡ f (incl c)
fib-f-l : (c : fst C (suc m)) → fib-f-incl (CW↪ C (suc m) c)
fst (fib-f-l c) = CW↪ D (suc m) (fm c)
snd (fib-f-l c) = sym (push _) ∙∙ ind c ∙∙ cong f (push _)
fib-f-r : (x : A (suc m))
→ fib-f-incl (invEq (e (suc m)) (inr x))
fib-f-r x = subst fib-f-incl (cong (invEq (e (suc m)))
(push (x , ptSn m)))
(fib-f-l (α (suc m) (x , ptSn m)))
fib-f-l-coh : (x : A (suc m))
→ PathP (λ i → fib-f-incl (invEq (e (suc m)) (push (x , ptSn m) i)))
(fib-f-l (α (suc m) (x , ptSn m)))
(fib-f-r x)
fib-f-l-coh x i =
transp (λ j → fib-f-incl (invEq (e (suc m)) (push (x , ptSn m) (i ∧ j))))
(~ i)
(fib-f-l (α (suc m) (x , ptSn m)))
mere-fib-f-coh : ∥ ((x : A (suc m)) (y : S₊ m)
→ PathP (λ i → fib-f-incl (invEq (e (suc m)) (push (x , y) i)))
(fib-f-l (α (suc m) (x , y)))
(fib-f-r x)) ∥₁
mere-fib-f-coh = invEq propTrunc≃Trunc1
(invEq (_ , InductiveFinSatAC 1 (card (suc m)) _)
λ a → fst propTrunc≃Trunc1 (sphereToTrunc (suc m)
(sphereElim' m
(λ x → isOfHLevelRetractFromIso m
(invIso (PathPIdTruncIso (suc m)))
(isOfHLevelPathP' m (isProp→isOfHLevelSuc m
(isContr→isProp
(isConnected-CW↪∞ (suc (suc m)) D _))) _ _))
∣ fib-f-l-coh a ∣ₕ)))
module _ (ind : ((c : fst C (suc m)) → incl (fm c) ≡ f (incl c)))
(ind2 : ((x : A (suc m)) (y : S₊ m)
→ PathP (λ i → fib-f-incl ind (invEq (e (suc m)) (push (x , y) i)))
(fib-f-l ind (α (suc m) (x , y)))
(fib-f-r ind x)))
where
toFiber : (c : fst C (suc (suc m)))
→ fiber (incl {n = (suc (suc m))}) (f (incl c))
toFiber = CWskel-elim C (suc m) (fib-f-l ind) (fib-f-r ind) ind2
toFiberβ : (c : fst C (suc m)) → toFiber (CW↪ C (suc m) c) ≡ fib-f-l ind c
toFiberβ = CWskel-elim-inl C (suc m) (fib-f-l ind) (fib-f-r ind) ind2
module _ (p : (n : Fin (suc (suc m)))
→ Σ[ h ∈ (fst C (fst n) → fst D (fst n)) ]
((c : fst C (fst n)) → incl (h c) ≡ f (incl c)))
(f' : (n : Fin (suc m)) (c : fst C (fst n))
→ p (fsuc n) .fst (CW↪ C (fst n) c) ≡
CW↪ D (fst n) (p (injectSuc n) .fst c))
(r : (n : A (suc m)) (y : S₊ m) →
CW↪ D (suc m) (p flast .fst (α (suc m) (n , y)))
≡ CW↪ D (suc m) (p flast .fst (α (suc m) (n , ptSn m))))
(ind : (n : Fin _) (y : S₊ m) →
PathP
(λ i →
fib-f-incl (p flast .fst) r (p flast .snd)
(invEq (e (suc m)) (push (n , y) i)))
(fib-f-l (p flast .fst) r (p flast .snd)
(α (suc m) (n , y)))
(fib-f-r (p flast .fst) r (p flast .snd) n)) where
FST-base : Σ[ h ∈ (fst C (suc (suc m)) → fst D (suc (suc m))) ]
((c : fst C (suc (suc m))) → incl (h c) ≡ f (incl c))
fst FST-base = fst ∘ toFiber _ _ _ ind
snd FST-base = snd ∘ toFiber _ _ _ ind
Goalₗ : (n : Fin (suc (suc (suc m)))) → Type _
Goalₗ n = Σ[ h ∈ (fst C (fst n) → fst D (fst n)) ]
((c : fst C (fst n)) → incl (h c) ≡ f (incl c))
mainₗ : (n : Fin (suc (suc (suc m)))) → Goalₗ n
mainₗ = elimFin FST-base p
mainᵣ : (n : Fin (suc (suc m))) (c : fst C (fst n))
→ mainₗ (fsuc n) .fst (CW↪ C (fst n) c)
≡ CW↪ D (fst n) (mainₗ (injectSuc n) .fst c)
mainᵣ = elimFin (λ c
→ funExt⁻ (cong fst (elimFinβ {A = Goalₗ} FST-base p .fst))
(CW↪ C (suc m) c)
∙ cong fst (toFiberβ _ _ _ ind c)
∙ cong (CW↪ D (suc m))
(sym (funExt⁻
(cong fst (elimFinβ {A = Goalₗ} FST-base p .snd flast)) c)))
λ x c → funExt⁻ (cong fst (elimFinβ {A = Goalₗ}
FST-base p .snd (fsuc x))) (CW↪ C (fst x) c)
∙ f' x c
∙ cong (CW↪ D (fst x))
(sym (funExt⁻ (cong fst
((elimFinβ {A = Goalₗ} FST-base p .snd) (injectSuc x))) c))
CWmap→finCellMap : (C : CWskel ℓ) (D : CWskel ℓ')
(f : realise C → realise D) (m : ℕ)
→ ∥ finCellApprox C D f m ∥₁
CWmap→finCellMap C D f m =
PT.map (λ {(g , hom)
→ finsequencemap (fst ∘ g) (λ r x → sym (hom r x))
, →FinSeqColimHomotopy _ _ (g flast .snd)})
(approx C D f m)
finCWmap→CellMap : ∀ {ℓ ℓ'} (n : ℕ) (C : finCWskel ℓ n) (D : CWskel ℓ')
(f : realise (finCWskel→CWskel n C) → realise D)
→ ∃[ ϕ ∈ cellMap (finCWskel→CWskel n C) D ]
realiseCellMap ϕ ≡ f
finCWmap→CellMap n C D f =
PT.map (λ {(ϕ , p) → ψ ϕ (funExt⁻ p)
, funExt λ x
→ subst (λ x → realiseCellMap (ψ ϕ (funExt⁻ p)) x ≡ f x)
(Iso.sec (converges→ColimIso
{seq = realiseSeq (finCWskel→CWskel n C)} n (C .snd .snd)) x)
(cong (incl {n = n})
(silly ϕ (funExt⁻ p) _)
∙ funExt⁻ p (fincl (n , (<ᵗsuc {n}))
(Iso.inv (converges→ColimIso n (C .snd .snd)) x)))})
(CWmap→finCellMap
(finCWskel→CWskel n C) D f n)
where
open SequenceMap renaming (map to smap)
open FinSequenceMap
module _ (ϕ : finCellMap n (finCWskel→CWskel n C) D)
(ϕid : (x : FinSeqColim n
(realiseSeq (finCWskel→CWskel n C)))
→ FinSeqColim→Colim n
(finCellMap→FinSeqColim (finCWskel→CWskel n C) D ϕ x)
≡ f (FinSeqColim→Colim n x)) where
C≃ : (k : ℕ) → fst C n ≃ fst C (k +ℕ n)
C≃ zero = idEquiv _
C≃ (suc k) = compEquiv (C≃ k) (_ , snd (snd C) k)
C→D : (k : ℕ) → fst C (k +ℕ n) → fst D (k +ℕ n)
C→D k = CW↪Iterate D n k
∘ fmap ϕ (n , <ᵗsuc {n})
∘ invEq (C≃ k)
C→D-cellular : (k : ℕ) (x : fst (finCWskel→CWskel n C) (k +ℕ n))
→ C→D (suc k) (CW↪ (finCWskel→CWskel n C) (k +ℕ n) x)
≡ CW↪ D (k +ℕ n) (C→D k x)
C→D-cellular k x =
cong (CW↪ D (k +ℕ n) ∘ CW↪Iterate D n k ∘ fmap ϕ (n , <ᵗsuc))
(cong (invEq (C≃ k)) (retEq (_ , snd (snd C) k) x))
mainlem : ∀ {ℓ ℓ'} (D : ℕ → Type ℓ) (C : ℕ → Type ℓ') (n : ℕ)
→ (C→D : (k : ℕ) → C (k +ℕ n) → D (k +ℕ n))
→ (D↑ : (n : ℕ) → D n → D (suc n))
→ (C↑ : (n : ℕ) → C n → C (suc n))
→ (cm : (k : ℕ) (x : _)
→ C→D (suc k) (C↑ (k +ℕ n) x)
≡ D↑ (k +ℕ n) (C→D k x))
(r : ℕ) (k : ℕ) (p4 : r +ℕ n ≡ k)
(x : C k) (z : ℕ) (p0 : suc r ≡ z) (p1 : suc k ≡ z +ℕ n)
(m : ℕ) (p3 : r +ℕ n ≡ m)
(p2 : z +ℕ n ≡ suc m)
→ D↑ m (subst D p3 (C→D r (subst C (sym p4) x)))
≡ subst D p2 (C→D z (subst C p1 (C↑ k x)))
mainlem C D n C→D D↑ C↑ cm r =
J> λ x → J> λ p1 → J> λ p2
→ cong (D↑ (r +ℕ n))
(transportRefl _ ∙ cong (C→D r) (transportRefl _))
∙ sym ((λ i → subst C (isSetℕ _ _ p2 refl i)
(C→D (suc r) (subst D (isSetℕ _ _ p1 refl i)
(C↑ (r +ℕ n) x))))
∙ transportRefl _
∙ cong (C→D (suc r)) (transportRefl _)
∙ cm r x)
ψm : (m : ℕ) → fst (finCWskel→CWskel n C) m → fst D m
ψm m with (Dichotomyℕ n m)
... | inl q = subst (fst D) (snd q)
∘ C→D (fst q)
∘ subst (fst C) (sym (snd q))
... | inr q = fmap ϕ (m , <ᵗ-trans (<→<ᵗ q) (<ᵗsuc {n}))
ψ : cellMap (finCWskel→CWskel n C) D
smap ψ = ψm
comm ψ m x with (Dichotomyℕ n m) | Dichotomyℕ n (suc m)
... | inl n≤m | inl n≤sucm =
mainlem (fst D) (fst C) n C→D (CW↪ D) (CW↪ (finCWskel→CWskel n C))
C→D-cellular _ _ _ _ _
(inj-+m {m = n} (cong suc (snd n≤m)
∙ sym (cong predℕ (cong suc (snd n≤sucm))))) _ _ _ _
... | inl n≤m | inr n>sucm =
⊥.rec (<-asym (<≤-trans n>sucm n≤m) (1 , refl))
... | inr n>m | inl (zero , n≤sucm) =
(cong (CW↪ D m)
(cong (λ p → fmap ϕ (m , p) x) (isProp<ᵗ _ _))
∙ fcomm ϕ (m , <→<ᵗ n>m) x)
∙ cong (λ p → fmap ϕ (suc m , p) c) (isProp<ᵗ _ _)
∙ λ j → transp (λ i → fst D (n≤sucm (i ∨ ~ j))) (~ j)
(fmap ϕ (n≤sucm (~ j) ,
isProp→PathP {B = λ j → n≤sucm (~ j) <ᵗ suc n}
(λ _ → isProp<ᵗ) (<→<ᵗ n>m) <ᵗsuc j)
(transp (λ i → fst C (n≤sucm (~ i ∨ ~ j))) (~ j)
c))
where
c = CW↪ (finCWskel→CWskel n C) m x
lem : n>m ≡ (0 , sym n≤sucm)
lem = isProp≤ _ _
... | inr n>m | inl (suc diff , n≤sucm) =
⊥.rec (<-asym (<≤-trans (diff , +-suc diff n ∙ n≤sucm) n>m) (0 , refl))
... | inr n>m | inr n>sucm =
cong (CW↪ D m)
(funExt⁻ (cong (fmap ϕ) (ΣPathP (refl , (isProp<ᵗ _ _)))) x)
∙ fcomm ϕ (m , <ᵗ-trans (<→<ᵗ n>sucm) <ᵗsuc) x
silly : (x : _) → smap ψ n x ≡ fmap ϕ (n , <ᵗsuc {n}) x
silly x with (Dichotomyℕ n n)
... | inl (zero , p) =
cong (λ p → subst (fst D) p (C→D zero (subst (fst C) (sym p) x)))
(isSetℕ _ _ p refl)
∙ transportRefl _
∙ cong (C→D zero) (transportRefl x)
... | inl (suc diff , p) =
⊥.rec (<ᵗ-irrefl {n} (<→<ᵗ (diff , (+-suc diff n ∙ p))))
... | inr p = ⊥.rec (<ᵗ-irrefl (<→<ᵗ p))
private
module SeqHomotopyTypes {ℓ ℓ'} {C : Sequence ℓ} {D : Sequence ℓ'} (m : ℕ)
(f-c g-c : FinSequenceMap (Sequence→FinSequence m C)
(Sequence→FinSequence m D))
where
f = fmap f-c
g = fmap g-c
f-hom = fcomm f-c
g-hom = fcomm g-c
cell-hom : (n : Fin (suc m)) (c : obj C (fst n)) → Type ℓ'
cell-hom n c = Sequence.map D (f n c) ≡ Sequence.map D (g n c)
cell-hom-coh : (n : Fin m) (c : obj C (fst n))
→ cell-hom (injectSuc n) c
→ cell-hom (fsuc n) (Sequence.map C c) → Type ℓ'
cell-hom-coh n c h h' =
Square (cong (Sequence.map D) h) h'
(cong (Sequence.map D) (f-hom n c))
(cong (Sequence.map D) (g-hom n c))
module approx {C : CWskel ℓ} {D : CWskel ℓ'}
(m : ℕ) (f-c g-c : finCellMap m C D)
(h∞' : FinSeqColim→Colim m ∘ finCellMap→FinSeqColim C D f-c
≡ FinSeqColim→Colim m ∘ finCellMap→FinSeqColim C D g-c) where
open SeqHomotopyTypes m f-c g-c
open CWskel-fields C
h∞ : (n : Fin (suc m)) (c : fst C (fst n))
→ Path (realise D) (incl (fmap f-c n c)) (incl (fmap g-c n c))
h∞ n c = funExt⁻ h∞' (fincl n c)
pathToCellularHomotopy₁ : (t : 1 <ᵗ suc m) (c : fst C 1)
→ ∃[ h ∈ cell-hom (1 , t) c ]
Square (h∞ (1 , t) c)
(cong incl h)
(push (f (1 , t) c))
(push (g (1 , t) c))
pathToCellularHomotopy₁ t c =
TR.rec squash₁
(λ {(d , p)
→ ∣ d
, (λ i j
→ hcomp (λ k →
λ {(i = i0) → doubleCompPath-filler
(sym (push (f (1 , t) c)))
(h∞ _ c)
(push (g (1 , t) c)) (~ k) j
; (i = i1) → incl (d j)
; (j = i0) → push (f (1 , t) c) (~ k ∨ i)
; (j = i1) → push (g (1 , t) c) (~ k ∨ i)})
(p (~ i) j)) ∣₁})
(isConnectedCong 1 (CW↪∞ D 2)
(isConnected-CW↪∞ 2 D) h∞* .fst)
where
h∞* : CW↪∞ D 2 (CW↪ D 1 (f (1 , t) c)) ≡ CW↪∞ D 2 (CW↪ D 1 (g (1 , t) c))
h∞* = sym (push (f (1 , t) c)) ∙∙ h∞ _ c ∙∙ push (g (1 , t) c)
pathToCellularHomotopy-ind : (n : Fin m)
→ (hₙ : (c : fst C (fst n)) → Σ[ h ∈ cell-hom (injectSuc n) c ]
(Square (h∞ (injectSuc n) c)
(cong incl h)
(push (f (injectSuc n) c))
(push (g (injectSuc n) c))))
→ ∃[ hₙ₊₁ ∈ ((c : fst C (suc (fst n)))
→ Σ[ h ∈ cell-hom (fsuc n) c ]
(Square (h∞ (fsuc n) c)
(cong incl h)
(push (f (fsuc n) c))
(push (g (fsuc n) c)))) ]
((c : _) → cell-hom-coh n c (hₙ c .fst)
(hₙ₊₁ (CW↪ C (fst n) c) .fst))
pathToCellularHomotopy-ind (zero , q) ind =
fst (propTrunc≃ (isoToEquiv (compIso (invIso rUnit×Iso)
(Σ-cong-iso-snd
λ _ → isContr→Iso isContrUnit
((λ x → ⊥.rec (CW₀-empty C x))
, λ t → funExt λ x → ⊥.rec (CW₀-empty C x))))))
(invEq propTrunc≃Trunc1
(invEq (_ , satAC-CW₁ 1 C _)
λ c → fst propTrunc≃Trunc1
(pathToCellularHomotopy₁ (fsuc (zero , q) .snd) c)))
pathToCellularHomotopy-ind (suc n' , w) ind =
PT.map (λ q → hₙ₊₁ q , hₙ₊₁-coh q) Πfiber-cong²-hₙ₊₁-push∞
where
n : Fin m
n = (suc n' , w)
Pushout→C = invEq (e (suc n'))
hₙ'-filler : (x : fst C (suc n')) → _
hₙ'-filler x =
doubleCompPath-filler
(sym (f-hom n x))
(ind x .fst)
(g-hom n x)
hₙ' : (x : fst C (suc n'))
→ f (fsuc n) (CW↪ C (suc n') x)
≡ g (fsuc n) (CW↪ C (suc n') x)
hₙ' x = hₙ'-filler x i1
hₙ₊₁-inl : (x : fst C (suc n'))
→ cell-hom (fsuc n) (invEq (e (suc n')) (inl x))
hₙ₊₁-inl x = cong (CW↪ D (suc (suc n'))) (hₙ' x)
h∞-push-coh : (x : fst C (suc n'))
→ Square (h∞ (injectSuc n) x) (h∞ (fsuc n) (CW↪ C (fst n) x))
(push (f (injectSuc n) x) ∙ (λ i₁ → incl (f-hom n x i₁)))
(push (g (injectSuc n) x) ∙ (λ i₁ → incl (g-hom n x i₁)))
h∞-push-coh x i j =
hcomp (λ k
→ λ {(i = i0) → h∞' j (fincl (injectSuc n) x)
; (i = i1) → h∞' j (fincl (fsuc n) (CW↪ C (fst n) x))
; (j = i0) → cong-∙ (FinSeqColim→Colim m)
(fpush n (f (injectSuc n) x))
(λ i₁ → fincl (fsuc n) (f-hom n x i₁)) k i
; (j = i1) → cong-∙ (FinSeqColim→Colim m)
(fpush n (g (injectSuc n) x))
(λ i₁ → fincl (fsuc n) (g-hom n x i₁)) k i})
(h∞' j (fpush n x i))
hₙ₊₁-inl-coh : (x : fst C (fst n))
→ Square (h∞ (fsuc n) (invEq (e (fst n)) (inl x)))
(cong incl (hₙ₊₁-inl x))
(push (f (fsuc n) (invEq (e (fst n)) (inl x))))
(push (g (fsuc n) (invEq (e (fst n)) (inl x))))
hₙ₊₁-inl-coh x i j =
hcomp (λ k
→ λ {(i = i0) → h∞ (fsuc n) (CW↪ C (fst n) x) j
; (i = i1) → push (hₙ' x j) k
; (j = i0) → push (f (fsuc n) (CW↪ C (fst n) x)) (k ∧ i)
; (j = i1) → push (g (fsuc n) (CW↪ C (fst n) x)) (k ∧ i)})
(hcomp (λ k
→ λ {(i = i0) → h∞-push-coh x k j
; (i = i1) → incl (hₙ'-filler x k j)
; (j = i0) → compPath-filler'
(push (f (injectSuc n) x))
((λ i₁ → incl (f-hom n x i₁))) (~ i) k
; (j = i1) → compPath-filler'
(push (g (injectSuc n) x))
((λ i₁ → incl (g-hom n x i₁))) (~ i) k})
(ind x .snd i j))
module _ (x : A (suc n')) (y : S₊ n') where
push-path-filler : I → I → Pushout (α (suc n')) fst
push-path-filler i j =
compPath-filler'' (push (x , y)) (sym (push (x , ptSn n'))) i j
push-path :
Path (Pushout (α (suc n')) fst)
(inl (α (suc n') (x , y)))
(inl (α (suc n') (x , ptSn n')))
push-path j = push-path-filler i1 j
D∞PushSquare : Type ℓ'
D∞PushSquare =
Square {A = realise D}
(cong (CW↪∞ D (suc (suc (suc n'))))
(hₙ₊₁-inl (snd C .snd .fst (suc n') (x , y))))
(cong (CW↪∞ D (suc (suc (suc n'))))
(hₙ₊₁-inl (snd C .snd .fst (suc n') (x , ptSn n'))))
(λ i → incl (CW↪ D (suc (suc n'))
(f (fsuc n) (Pushout→C (push-path i)))))
(λ i → incl (CW↪ D (suc (suc n'))
(g (fsuc n) (Pushout→C (push-path i)))))
cong² : PathP (λ i → cell-hom (fsuc n)
(Pushout→C (push-path i)))
(hₙ₊₁-inl (α (suc n') (x , y)))
(hₙ₊₁-inl (α (suc n') (x , ptSn n')))
→ D∞PushSquare
cong² p i j = incl (p i j)
isConnected-cong² : isConnectedFun (suc n') cong²
isConnected-cong² =
isConnectedCong² (suc n') (CW↪∞ D (3 +ℕ n'))
(isConnected-CW↪∞ (3 +ℕ n') D)
hₙ₊₁-push∞ : D∞PushSquare
hₙ₊₁-push∞ i j =
hcomp (λ k
→ λ {(i = i0) → hₙ₊₁-inl-coh (α (suc n') (x , y)) k j
; (i = i1) → hₙ₊₁-inl-coh (α (suc n') (x , ptSn n')) k j
; (j = i0) → push (f (fsuc n) (Pushout→C (push-path i))) k
; (j = i1) → push (g (fsuc n) (Pushout→C (push-path i))) k})
(hcomp (λ k
→ λ {(i = i0) → h∞' j (fincl (fsuc n)
(Pushout→C (push (x , y) (~ k))))
; (i = i1) → h∞' j (fincl (fsuc n)
(Pushout→C (push (x , ptSn n') (~ k))))
; (j = i0) → incl (f (fsuc n)
(Pushout→C (push-path-filler k i)))
; (j = i1) → incl (g (fsuc n)
(Pushout→C (push-path-filler k i)))})
(h∞' j (fincl (fsuc n) (Pushout→C (inr x)))))
fiber-cong²-hₙ₊₁-push∞ : hLevelTrunc (suc n') (fiber cong² hₙ₊₁-push∞)
fiber-cong²-hₙ₊₁-push∞ = isConnected-cong² hₙ₊₁-push∞ .fst
hₙ₊₁-coh∞ : (q : fiber cong² hₙ₊₁-push∞)
→ Cube (hₙ₊₁-inl-coh (α (suc n') (x , y)))
(hₙ₊₁-inl-coh (α (suc n') (x , ptSn n')))
(λ j k → h∞' k (fincl (fsuc n) (Pushout→C (push-path j))))
(λ j k → incl (fst q j k))
(λ j i → push (f (fsuc n) (Pushout→C (push-path j))) i)
λ j i → push (g (fsuc n) (Pushout→C (push-path j))) i
hₙ₊₁-coh∞ q j i k =
hcomp (λ r
→ λ {(i = i0) → h∞' k (fincl (fsuc n) (Pushout→C (push-path j)))
; (i = i1) → q .snd (~ r) j k
; (j = i0) → hₙ₊₁-inl-coh (α (suc n') (x , y)) i k
; (j = i1) → hₙ₊₁-inl-coh (α (suc n') (x , ptSn n')) i k
; (k = i0) → push (f (fsuc n) (Pushout→C (push-path j))) i
; (k = i1) → push (g (fsuc n) (Pushout→C (push-path j))) i})
(hcomp (λ r
→ λ {(i = i0) → h∞' k (fincl (fsuc n) (Pushout→C (push-path j)))
; (j = i0) → hₙ₊₁-inl-coh (α (suc n') (x , y)) (i ∧ r) k
; (j = i1) → hₙ₊₁-inl-coh (α (suc n') (x , ptSn n')) (i ∧ r) k
; (k = i0) → push (f (fsuc n)
(Pushout→C (push-path j))) (i ∧ r)
; (k = i1) → push (g (fsuc n)
(Pushout→C (push-path j))) (i ∧ r)})
(hcomp (λ r
→ λ {(i = i0) → h∞' k (fincl (fsuc n)
(Pushout→C (push-path-filler r j)))
; (j = i0) → h∞' k (fincl (fsuc n) (invEq (e (suc n'))
(push (x , y) (~ r))))
; (j = i1) → h∞' k (fincl (fsuc n) (invEq (e (suc n'))
(push (x , ptSn n') (~ r))))
; (k = i0) → incl (f (fsuc n)
(Pushout→C (push-path-filler r j)))
; (k = i1) → incl (g (fsuc n)
(Pushout→C (push-path-filler r j)))})
(h∞' k (fincl (fsuc n) (Pushout→C (inr x))))))
Πfiber-cong²-hₙ₊₁-push∞ :
∥ ((x : _) (y : _) → fiber (cong² x y) (hₙ₊₁-push∞ x y)) ∥₁
Πfiber-cong²-hₙ₊₁-push∞ =
Iso.inv propTruncTrunc1Iso
(invEq (_ , InductiveFinSatAC 1 _ _)
λ x → Iso.fun propTruncTrunc1Iso
(sphereToTrunc (suc n') (fiber-cong²-hₙ₊₁-push∞ x)))
module _ (q : (x : Fin (fst (snd C) (suc n'))) (y : S₊ n')
→ fiber (cong² x y) (hₙ₊₁-push∞ x y)) where
agrees : (x : fst C (suc n')) (ϕ : cell-hom (fsuc n) (CW↪ C (suc n') x))
→ Type _
agrees x ϕ = Square (h∞ (fsuc n) (CW↪ C (suc n') x))
(cong incl ϕ)
(push (f (fsuc n) (CW↪ C (suc n') x)))
(push (g (fsuc n) (CW↪ C (suc n') x)))
main-inl : (x : fst C (suc n'))
→ Σ (cell-hom (fsuc n) (CW↪ C (suc n') x))
(agrees x)
main-inl x = hₙ₊₁-inl x , hₙ₊₁-inl-coh x
main-push : (x : A (suc n')) (y : S₊ n')
→ PathP (λ i → Σ[ ϕ ∈ (cell-hom (fsuc n) (Pushout→C (push-path x y i))) ]
(Square (h∞ (fsuc n) (Pushout→C (push-path x y i)))
(λ j → incl (ϕ j))
(push (f (fsuc n) (Pushout→C (push-path x y i))))
(push (g (fsuc n) (Pushout→C (push-path x y i))))))
(main-inl (α (suc n') (x , y)))
(main-inl (α (suc n') (x , ptSn n')))
main-push x y = ΣPathP (fst (q x y) , hₙ₊₁-coh∞ x y (q x y))
hₙ₊₁ : (c : fst C (fst (fsuc n)))
→ Σ[ ϕ ∈ (cell-hom (fsuc n) c) ]
Square (h∞ (fsuc n) c)
(cong incl ϕ)
(push (f (fsuc n) c))
(push (g (fsuc n) c))
hₙ₊₁ = CWskel-elim' C n' main-inl main-push
hₙ₊₁-coh-pre : (c : fst C (suc n')) →
Square (cong (CW↪ D (suc (suc n'))) (ind c .fst))
(hₙ₊₁-inl c)
(cong (CW↪ D (suc (suc n'))) (f-hom n c))
(cong (CW↪ D (suc (suc n'))) (g-hom n c))
hₙ₊₁-coh-pre c i j = CW↪ D (suc (suc n')) (hₙ'-filler c i j)
hₙ₊₁-coh : (c : fst C (suc n')) →
Square (cong (CW↪ D (suc (suc n'))) (ind c .fst))
(hₙ₊₁ (CW↪ C (suc n') c) .fst)
(cong (CW↪ D (suc (suc n'))) (f-hom n c))
(cong (CW↪ D (suc (suc n'))) (g-hom n c))
hₙ₊₁-coh c = hₙ₊₁-coh-pre c
▷ λ i → CWskel-elim'-inl C n'
{B = λ c → Σ[ ϕ ∈ cell-hom (fsuc n) c ]
Square (h∞ (fsuc n) c) (cong incl ϕ)
(push (f (fsuc n) c)) (push (g (fsuc n) c))}
main-inl main-push c (~ i) .fst
private
pathToCellularHomotopy-main :
{C : CWskel ℓ} {D : CWskel ℓ'} (m : ℕ) (f-c g-c : finCellMap m C D)
(h∞' : FinSeqColim→Colim m ∘ finCellMap→FinSeqColim C D f-c
≡ FinSeqColim→Colim m ∘ finCellMap→FinSeqColim C D g-c)
→ ∥ finCellHomRel m f-c g-c (approx.h∞ m f-c g-c h∞') ∥₁
pathToCellularHomotopy-main {C = C} zero f-c g-c h∞' =
∣ fincellhom (λ {(zero , p) x → ⊥.rec (CW₀-empty C x)})
(λ { (zero , p) x → ⊥.rec (CW₀-empty C x)})
, (λ { (zero , p) x → ⊥.rec (CW₀-empty C x)}) ∣₁
pathToCellularHomotopy-main {C = C} {D = D} (suc zero) f-c g-c h∞' =
PT.map (λ {(d , h)
→ (fincellhom (λ {(zero , p) x → ⊥.rec (CW₀-empty C x)
; (suc zero , p) → d})
λ {(zero , p) → λ x → ⊥.rec (CW₀-empty C x)})
, (λ {(zero , p) x → ⊥.rec (CW₀-empty C x)
; (suc zero , tt) → h})})
(invEq (_ , satAC∃Fin-C0 C _ _) k)
where
k : (c : _) → _
k c = (approx.pathToCellularHomotopy₁ (suc zero) f-c g-c
h∞' tt c)
pathToCellularHomotopy-main {C = C} {D = D} (suc (suc m)) f-c g-c h∞' =
PT.rec squash₁
(λ ind → PT.map
(λ {(f , p)
→ (fincellhom (main-hom ind f p)
(main-coh ind f p))
, (∞coh ind f p)})
(pathToCellularHomotopy-ind flast
λ c → (finCellHom.fhom (ind .fst) flast c)
, (ind .snd flast c)))
(pathToCellularHomotopy-main {C = C} {D = D} (suc m)
(finCellMap↓ f-c)
(finCellMap↓ g-c)
h')
where
open approx _ f-c g-c h∞'
finSeqColim↑ : ∀ {ℓ} {X : Sequence ℓ} {m : ℕ}
→ FinSeqColim m X → FinSeqColim (suc m) X
finSeqColim↑ (fincl n x) = fincl (injectSuc n) x
finSeqColim↑ {m = suc m} (fpush n x i) = fpush (injectSuc n) x i
h' : FinSeqColim→Colim (suc m) ∘
finCellMap→FinSeqColim C D (finCellMap↓ f-c)
≡
FinSeqColim→Colim (suc m) ∘
finCellMap→FinSeqColim C D (finCellMap↓ g-c)
h' = funExt λ { (fincl n x) j → h∞' j (fincl (injectSuc n) x)
; (fpush n x i) j → h∞' j (fpush (injectSuc n) x i)}
open SeqHomotopyTypes
module _
(ind : finCellHomRel (suc m)
(finCellMap↓ f-c) (finCellMap↓ g-c)
((approx.h∞ (suc m) (finCellMap↓ f-c) (finCellMap↓ g-c) h')))
(f : (c : fst C (suc (suc m))) →
Σ[ h ∈ (cell-hom (suc (suc m)) f-c g-c flast c) ]
(Square (h∞ flast c) (cong incl h)
(push (fmap f-c flast c)) (push (fmap g-c flast c))))
(fp : (c : fst C (suc m)) →
cell-hom-coh (suc (suc m)) f-c g-c flast c
(finCellHom.fhom (ind .fst) flast c)
(f (CW↪ C (suc m) c) .fst)) where
main-hom-typ : (n : Fin (suc (suc (suc m))))
→ Type _
main-hom-typ n =
(x : C .fst (fst n))
→ CW↪ D (fst n) (f-c .fmap n x)
≡ CW↪ D (fst n) (g-c .fmap n x)
main-hom : (n : Fin (suc (suc (suc m)))) → main-hom-typ n
main-hom = elimFin (fst ∘ f) (finCellHom.fhom (fst ind))
main-homβ = elimFinβ {A = main-hom-typ} (fst ∘ f)
(finCellHom.fhom (fst ind))
main-coh : (n : Fin (suc (suc m))) (c : C .fst (fst n))
→ Square
(cong (CW↪ D (suc (fst n)))
(main-hom (injectSuc n) c))
(main-hom (fsuc n)
(CW↪ C (fst n) c))
(cong (CW↪ D (suc (fst n))) (fcomm f-c n c))
(cong (CW↪ D (suc (fst n))) (fcomm g-c n c))
main-coh =
elimFin
(λ c → cong (cong (CW↪ D (suc (suc m))))
(funExt⁻ (main-homβ .snd flast) c)
◁ fp c
▷ sym (funExt⁻ (main-homβ .fst) (CW↪ C (suc m) c)))
λ n c
→ cong (cong (CW↪ D (suc (fst n))))
(funExt⁻ (main-homβ .snd (injectSuc n)) c)
◁ finCellHom.fcoh (fst ind) n c
▷ sym (funExt⁻ (main-homβ .snd (fsuc n)) (CW↪ C (fst n) c))
∞coh : (n : Fin (suc (suc (suc m)))) (x : fst C (fst n))
→ Square (h∞ n x) (λ i → incl (main-hom n x i))
(push (f-c .fmap n x)) (push (g-c .fmap n x))
∞coh = elimFin
(λ c → f c .snd ▷ λ i j → incl (main-homβ .fst (~ i) c j))
λ n c → ind .snd n c ▷ λ i j → incl (main-homβ .snd n (~ i) c j)
pathToCellularHomotopy :
{C : CWskel ℓ} {D : CWskel ℓ'} (m : ℕ) (f-c g-c : finCellMap m C D)
→ ((x : fst C m) → Path (realise D) (incl (fmap f-c flast x))
(incl (fmap g-c flast x)))
→ ∥ finCellHom m f-c g-c ∥₁
pathToCellularHomotopy {C} {D} m f-c g-c h =
PT.map fst
(pathToCellularHomotopy-main m f-c g-c
(→FinSeqColimHomotopy _ _ h))